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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Chebychev design of an FIR filter given a desired $H(\\omega)$\n",
      "\n",
      "A derivative work by Judson Wilson, 5/27/2014.<br>\n",
      "Adapted from the CVX example of the same name, by Almir Mutapcic, 2/2/2006.\n",
      "\n",
      "Topic References:\n",
      "\n",
      "* \"Filter design\" lecture notes (EE364) by S. Boyd\n",
      "\n",
      "## Introduction\n",
      "\n",
      "This program designs an FIR filter, given a desired frequency response $H_\\mbox{des}(\\omega)$.\n",
      "The design is judged by the maximum absolute error (Chebychev norm).\n",
      "This is a convex problem (after sampling it can be formulated as an SOCP),\n",
      "which may be written in the form:\n",
      "    \\begin{array}{ll}\n",
      "    \\mbox{minimize}   &  \\max |H(\\omega) - H_\\mbox{des}(\\omega)| \n",
      "                             \\quad \\mbox{ for }  0 \\le \\omega \\le \\pi,\n",
      "    \\end{array}\n",
      "where the variable $H$ is the frequency response function, corresponding to an impulse response $h$.\n",
      "\n",
      "## Initialize problem data"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import cvxpy as cvx\n",
      "\n",
      "#********************************************************************\n",
      "# Problem specs.\n",
      "#********************************************************************\n",
      "# Number of FIR coefficients (including the zeroth one).\n",
      "n = 20\n",
      "\n",
      "# Rule-of-thumb frequency discretization (Cheney's Approx. Theory book).\n",
      "m = 15*n\n",
      "w = np.mat(np.linspace(0,np.pi,m)).T\n",
      "\n",
      "#********************************************************************\n",
      "# Construct the desired filter.\n",
      "#********************************************************************\n",
      "# Fractional delay.\n",
      "D = 8.25                # Delay value.\n",
      "Hdes = np.exp(-1j*D*w)  # Desired frequency response.\n",
      "\n",
      "# Gaussian filter with linear phase. (Uncomment lines below for this design.)\n",
      "#var = 0.05\n",
      "#Hdes = 1/(np.sqrt(2*np.pi*var)) * np.exp(-np.square(w-np.pi/2)/(2*var))\n",
      "#Hdes = np.multiply(Hdes, np.exp(-1j*n/2*w))\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Solve the minimax (Chebychev) design problem"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# A is the matrix used to compute the frequency response\n",
      "# from a vector of filter coefficients:\n",
      "#     A[w,:] = [1 exp(-j*w) exp(-j*2*w) ... exp(-j*n*w)]\n",
      "A = np.exp( -1j * np.kron(np.mat(w), np.mat(np.arange(n))) )\n",
      "\n",
      "# Presently CVXPY does not do complex-valued math, so the\n",
      "# problem must be formatted into a real-valued representation.\n",
      "\n",
      "# Split Hdes into a real part, and an imaginary part.\n",
      "Hdes_r = np.real(Hdes)\n",
      "Hdes_i = np.imag(Hdes)\n",
      "\n",
      "# Split A into a real part, and an imaginary part.\n",
      "A_R = np.real(A)\n",
      "A_I = np.imag(A)\n",
      "\n",
      "#\n",
      "# Optimal Chebyshev filter formulation.\n",
      "#\n",
      "\n",
      "# h is the (real) FIR coefficient vector, which we are solving for.\n",
      "h = cvx.Variable(shape=(n,1))\n",
      "# The objective is:\n",
      "#     minimize max(|A*h-Hdes|)\n",
      "# but modified into an equivelent form:\n",
      "#     minimize max( real(A*h-Hdes)^2 + imag(A*h-Hdes)^2 )\n",
      "# such that all computation is done in real quantities only.\n",
      "obj = cvx.Minimize(\n",
      "        cvx.max( \n",
      "           cvx.square(A_R * h - Hdes_r)     # Real part.\n",
      "         + cvx.square(A_I * h - Hdes_i) ) ) # Imaginary part.\n",
      "\n",
      "# Solve problem.\n",
      "prob = cvx.Problem(obj)\n",
      "prob.solve()\n",
      "\n",
      "# Check if problem was successfully solved.\n",
      "print 'Problem status:', prob.status\n",
      "if prob.status != cvx.OPTIMAL:\n",
      "    raise Exception('CVXPY Error')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Problem status: optimal\n"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Result plots"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import matplotlib.pyplot as plt\n",
      "\n",
      "# Show plot inline in ipython.\n",
      "%matplotlib inline\n",
      "\n",
      "# Plot properties.\n",
      "plt.rc('text', usetex=True)\n",
      "plt.rc('font', family='serif')\n",
      "font = {'family' : 'normal',\n",
      "        'weight' : 'normal',\n",
      "        'size'   : 16}\n",
      "plt.rc('font', **font)\n",
      "\n",
      "# Plot the FIR impulse reponse.\n",
      "plt.figure(figsize=(6, 6))\n",
      "plt.stem(range(n),h.value)\n",
      "plt.xlabel('n')\n",
      "plt.ylabel('h(n)')\n",
      "plt.title('FIR filter impulse response')\n",
      "\n",
      "# Plot the frequency response.\n",
      "H = np.exp(-1j * np.kron(w, np.mat(np.arange(n)))) * h.value\n",
      "plt.figure(figsize=(6, 6))\n",
      "# Magnitude\n",
      "plt.plot(np.array(w), 20 * np.log10(np.array(np.abs(H))),\n",
      "         label='optimized')\n",
      "plt.plot(np.array(w), 20 * np.log10(np.array(np.abs(Hdes))),'--',\n",
      "         label='desired')\n",
      "plt.xlabel(r'$\\omega$')\n",
      "plt.ylabel(r'$|H(\\omega)|$ in dB')\n",
      "plt.title('FIR filter freq. response magnitude')\n",
      "plt.xlim(0, np.pi)\n",
      "plt.ylim(-30, 10)\n",
      "plt.legend(loc='lower right')\n",
      "# Phase\n",
      "plt.figure(figsize=(6, 6))\n",
      "plt.plot(np.array(w), np.angle(np.array(H)))\n",
      "plt.xlim(0, np.pi)\n",
      "plt.ylim(-np.pi, np.pi)\n",
      "plt.xlabel(r'$\\omega$')\n",
      "plt.ylabel(r'$\\angle H(\\omega)$')\n",
      "plt.title('FIR filter freq. response angle')\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 4,
       "text": [
        "<matplotlib.text.Text at 0x11125d410>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
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       "text": [
        "<matplotlib.figure.Figure at 0x1110959d0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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SfpnxJQmZoLKzfHIAQDuKEkjOKj++5KTKP6gKANCmNlMd1aHiJxgmJJ2WdFPk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\naGjItizLTiQS9ujoqD09PW2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       "text": [
        "<matplotlib.figure.Figure at 0x111219550>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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l3E+EomRcFtCqDD+59uHOnXD+vJv7m8J1a74Kw08u43DKFLWbcVnlWYTCAS67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\nKGIKrizcdJPaa6fT0In4rzVZDjISHzZ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       "text": [
        "<matplotlib.figure.Figure at 0x1113122d0>"
       ]
      }
     ],
     "prompt_number": 4
    }
   ],
   "metadata": {}
  }
 ]
}